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An inverse technique to deduce the elasticity of a
large artery.
Pierre-Yves Lagr�eeLaboratoire de Mod�elisation en M�ecanique,
Universit�e Paris VI,UMR 7607
Bo�te 162, 4 place Jussieu,75005 PARIS FRANCE.
Revision 2 of manuscript AP9016 (EPJ-AP).Pacs Numbers: 47.15-x, 47.60+i, 87.22-q
17th January 2000
Abstract
Our purpose is to build an inverse method which best �ts a model ofartery ow and experimental measurements (we assume that we are able tomeasure the displacement of the artery as a function of time at three sta-tions). Having no clinical data, we simulate these measurements with thenumerical computations from a \boundary layer" code. First, we revisitthe system of Ling & Atabek of boundary layer type for the transmissionof a pressure pulse in the arterial system for the case of an elastic wall (butwe solve it without any simpli�cation in the u@u=@x term). Then, usinga method analogous to the well known Von K�arm�an-Pohlhausen methodfrom aeronautics but transposed here for a pulsatile ow, we build a sys-tem of three coupled non-linear partial di�erential equations dependingonly on time and axial co-ordinate. This system governs the dynamics ofinternal artery radius, centre velocity and a quantity related to the pres-ence of viscous e�ects. These two methods give nearly the same numericalresults.
Second, we construct an inverse method: the aim is to �nd for thesimple integral model, the physical parameters to put in the \boundarylayer" code (simulating clinical data). This is done by varying in the inte-gral model the viscosity and elasticity in order to �t best with the data. Toachieve this in a rational way, we have to minimise a cost function, whichinvolves the computation of the adjoint system of the integral method.The good set of parameters (i :e: viscosity, and two coeÆcients of a walllaw) is e�ectively found again. It opens the perspective for applicationin real clinical cases of this new non-invasive method for evaluating theviscosity of the ow and elasticity of the wall.
1
1 Nomenclature
.u longitudinal velocityv transversal velocityh displacement of the artery radiusp pressurex longitudinal variabler radial transversal variable� reduced variable r=Rt timeR artery radiusR0 unperturbed artery radius� Womersley number! pulsationk elasticity of the wall�k1, �k2 adimensionalized coeÆcient of elasticityU0 velocity at the center of the pipeQ the uxq defect of ux: an integral linked with u� an integral linked with u2
� coeÆcients of skin friction coeÆcients of �ÆR the gauge of hL the longitudinal scale"1 a small parameter R=L"2 a parameter ÆR=R"n amplitude of random noiseJ the cost functionJ0 Bessel function of order 0J1 Bessel function of order 1� adjoint variablesL the Lagrangian
2 Introduction
Numerical computation of blood propagation in arteries is a great challenge formechanics: it involves an unsteady 3D motion of a complex non-homogenousnon-Newtonian uid interacting with a moving changing shape 3D geometry.Most of the physical parameters involved are unknown or diÆcult to mea-sure, numerical methods are under development but computers are not pow-erful enough. Mathematical settlements of convergence of the process are notcompletely established (Errate et al. (1994) [7]). Nevertheless, many examplesof successful computations with simple descriptions of the uid and/ or the wallcan be found in the literature. Among them, Reuderink et al (1993) [26] use thelinear wave propagation of the small disturbance as boundary conditions for a
2
full Newtonian Navier Stokes computation, the input being a sinusoidal pulse(to compare with Womersley (1955) [38] results). Vesier & Yoganathan (1992)[35] present a real interaction between a full Newtonian axial- symmetric NavierStokes code and an elastic wall (string law and no inertia), the input being asinusoidal pulse again. Ma, Lee & Wu (1992) [17] (�rst models are in Wu, Lee& Tseng (1984) [17] and Wu & Lee (1989) [17]) present a computation withfull Newtonian axi- symmetrical Navier Stokes code with a re�ned descriptionof the wall displacement (though �nally linearized). E�ects of cross-sectionchange with a rigid wall are computed by Pedrizzetti (1996) [22].
When the phenomena in the wall were emphasised (inertia, axial and lon-gitudinal tension, viscoelasticity etc) the ow was simpli�ed. The velocity issupposed to be constant over the section and the viscous e�ect is a frictionforce (Seymour (1975) [30] studied the competition between dissipation andnon-linear e�ects in the propagation of a monochromatic ow). The ow isoften even linearized as in Kuiken (1984) [10] though it remains viscous, or it istaken as a perfect uid though it remains non-linear as in Yomosa (1986) [42] orin Paquerot & Remoissenet (1994) [21] (in the two last references non-linearityand dispersion lead to solitons).
Nevertheless, most of the operational techniques use a simpli�ed set of equa-tions for the uid and for the wall: the equations are averaged on the section andthe skin friction (which appears during the integration over the section) is mod-eled by a "closure" relation involving the ux and the viscosity (deduced fromthe guessed velocity pro�le). See for example Pedley (1980) [23], Zagzoule &Marc- Vergnes (1986) [44], Horsten et al. (1989) [12], Zagzoule, Khalid- Naciri& Mauss (1991)[43], Mederic, Gaudu, Mauss & Zagzoule (1991) [18], Yama,Mederic & Zagzoule (1995) [41], Rogova & Flaud (1995) [27], Pythoud Ster-giopulos & Meister (1996) [25] or to a certain extent Belardinelli & Cavalcanti(1992) [2].
These methods, based on drastic simpli�cations of the complete equationssystem, are powerful in large arteries. This is due to the fact that the domi-nant mechanism, as explained by Lighthill (1975) [15] and Pedley (1980) [23],involves the propagation of a rather small amplitude pressure wave and thatthe viscosity and the viscoelasticity are not particularly dissipative. So, herewe adopt a simple point of view and simplify the full Navier Stokes equations;we use a kind of boundary layer (or thin layer) approach of the equations andwork with the Ling & Atabek (1972) [16] system (referred as "boundary layersystem"). Simplifying much more, in the spirit of the preceding references, wewill integrate these equations transversally and use a closure from the Wom-ersley (1955) [38] linear solution (the result is referred as "integral system").Comparisons between numerical simulations of the �nal integral system and ofthe original boundary layer system (without any simpli�cation such as the kfunction of Ling & Atabek (1972) [16]) are exposed, and we shall see that theyare similar in linear and non-linear cases.
The main reason for the interest in the ow in an elastic pipe is the beliefthat it may help to understand a pathological state of an artery. In this scope,an interesting tool would be to measure the elasticity (or the compliance) of thewall in vivo, in a non-invasive way to predict or follow the evolution of diseases
3
(or we may want to study how to develop a prosthesis for a femoral artery). Theidea is that we have a simpli�ed model (depending on a set of coeÆcients, in thiscase the viscosity and the linear compliance of the wall, but other parametersmay be put in the model) and we want the set of coeÆcients which describesbest the experiments. Our aim is di�erent from Rogova & Flaud's (1995) [27]and from Pythoud Stergiopulos & Meister's (1996) [25], who are interestedin separating the forward and backward waves, but we also use a simpli�edintegral model for the ow. We suppose that it is possible (we do not discussthe experimental diÆculties and precision problem) to obtain from Dopplermeasurements the displacement (or the pressure) at three di�erent locationsxin < xm < xout, as a function of time (during a period). Two preceding papers([27], and [25]) need measurements of pressure and velocity at one location. Themeasured displacements hin(t) and hout(t) are then the boundary conditions forthe simpli�ed integral model, hm(t) is compared to the numerically computedresult (from the integral model) of the displacement obtained in xm; the aimis to �nd out the set of parameters which makes the measurement and thecomputation the closest. The so called "inverse method" (Chavent (1979) [3])allows us to construct eÆciently such a method. Therefore, we build a costfunction which has to be minimised with the help of the numerical resolutionof the integral system's adjoint system which allows us to compute the gradientof this cost function.
These methods of back propagation were �rst developed in seismographyin order to "guess" the internal structure of the earth and the location of anearthquake knowing only the seismic measurements (Tarantola (1987) [31]).More generally, any complex system may be modelled by a more simple setof coupled O.D.E. or P.D.E. depending on a set of parameters, the inversemethod leading to the set of optimum parameters. Among others, we mayquote the interaction of the wakes (K�arm�an streets) which are issued from arow of cylinders (Fullana et al. [9] (1997)), or Barros (1995) [1] where thecoeÆcient of friction at the river bed is deduced from measurements of thewater level. Having up to now no experimental results, we shall use our thinlayer resolution of the Ling & Atabek's system as the "experiment", and ourintegral description as the model to adjust. So we shall show that the optimumset of parameters found for the integral method are not too far from the "real"ones put in the thin layer resolution, opening the way to future more complexmodelling and to real comparisons with clinical data.
3 Dynamical model
3.1 Analysis
An homogenous, Newtonian, incompressible (with constant density �) uidis assumed, the viscosity � is constant (� = �=�); these hypotheses may berelevant for blood if the arteries are wide enough and if we assume that the non-Newtonian behaviour (viscoelasticity) and the fact that blood is a suspensionof cells (deformable solid objects) are negligible. To a certain extent � and �may be taken as re-normalised values (Flaud et Quemada, (1980) [8]). The ow
4
is assumed to be axisymmetrical, gravity e�ects are neglected. It is describedby the longitudinal u(r; x; t) and the radial v(r; x; t) components of the velocity~u. The pressure in the uid is p(r; x; t). The problem is then to solve theNavier-Stokes equations (for the uid):
~r � ~u = 0 &@~u
@t+ ~u � ~r~u = �
~rp�
+ � ~r2~u: (1)
This set of equations is coupled with the artery motion equations (we do notwrite the general equations for the solid, called the "wall"). The most generalboundary conditions are the equality of the velocities at the wall (no slip con-dition) and the equality of the stresses at the wall. This set of equations hasbeen solved by Vesier & Yoganathan (1992) [35] or Ma, Lee & Wu (1992) [17](some mathematical diÆculties were shown by Errate et al [7]). But here ourscope is to obtain a simpli�ed system for quick resolution with small numericalfacilities.
In �gure 1, we present a rough sketch of the notations. As usual, we intro-duce by phenomenological analysis, small parameters to simplify the equations.First, the no slip condition at the wall r = R(x; t) is simpli�ed as follows:
v(r; x; t)jr=R =@R
@t; u(r; x; t)jr=R = 0; (2)
This dictates only radial movements: this is not the most general boundarycondition (Womersley (1955) [38] retains the two displacements). The wall istethered to its sti�er surroundings: it seems to be very diÆcult in practice, tomeasure this displacement. The pressure is scaled by the elasticity of the wall:if the wall is moved by an amount ÆR from the equilibrium radius R0 (roughlyabout 0:5cm; and ÆR=R0 is much smaller than 0:1), the simplest expression forthe restoring pressure is k(ÆR) (roughly about 13kPa). With the same scaleÆR for the variations of the radius and with T the period of blood ejection(characteristic time T ' 1; 2s), we scale the transverse velocity by ÆR=T . GivenL, a characteristic length, and u0, a characteristic longitudinal velocity (up to
Figure 1: the ow in the elastic pipe.
5
now unknown), and noticing that transverse variations are scaled by R0; theincompressibility imposes u0 � L(ÆR=R0)=T , and the longitudinal momentum
�u0=T � k(ÆR)=L: Hence a good possible choice for L is Tq
kR0
� and for u0
ÆRR0
qkR0
� (note c0 =q
12
qkR0
� is known as the Moens- Korteweg celerity and
the value is about 3 to 10m=s), other good choices for scales are linked to thisone (except factors � or
p2).
The preceding scales de�ne an adimensionalized parameter that we maycall "2 = ÆR
R0. It measures the importance of the non-linearity in the uid
((~u � ~r) =(@@t ) = O("2)) as well as in the wall. This parameter is typically
much smaller than 0:1. The ratio "1 = R0
L is in practice smaller than 10�2,enabling the transverse variation of pressure to be neglected, which is of theorder of "21: Finally, the relative importance of viscosity is related to the ratioof the viscous term by the unsteady term: (�u0=R
2)=(u0=T ). This permits us
to de�ne the classical Womersley number � = R0
q2�=T� . The bigger �, the
atter the velocity pro�le; the smaller �, the better we obtain a Hagen-Poiseuille ow. We shall consider that in practice (for femoral arteries) � is between 3and 5. The longitudinal derivative viscous term is of course "21 smaller than thetransversal one. Finally, we must mention the Reynolds number, though for thisunsteady ow the Womersley number is more relevant. The Reynolds numberconstructed with the diameter is ReD = (u0(2R0))=�. After substitution this isReD = ��1"2"
�11 �2 which is at most 100.
3.2 Final form for the uid
With the following adimensionalization:
x = Tq
(kR0)� �x; r = R0�r; t = T �t
u = "2
q(kR0)� �u; v = "2
R0
T �v; p = "2(kR0)�p; R = R0�R:
where c0 =q
12kR0
� and "1 = R0
L , "2 = ÆRR0
and � = R0
q2�=T� ; we write
again the system (1 ). As the boundary conditions are evaluated on a movingunknown surface (the location of the wall is �r = �R(x; t)) we map the equationsin introducing a new variable: �� = �r= �R(�x; �t). The boundary conditions arethen on a �xed surface: �� = 1, but it introduces new terms in the equations(e.g. the @=@�t term is now @=@�t� (��= �R)(@ �R=@�t)@=@�� and so on). As "1 is thereal small parameter ("2 is not necessarily small, and � not necessarily large),we obtain a boundary layer like system of equations:
�R = 1 + "2�h
1
R
@�v
@��+
�v
�� �R+@�u
@�x� "2 � ���R
@ �R
@�x
@�u
@��= 0 ; (3)
@�u
@�t� "2 � ���R � @
�h
@�t� @�u@��
+ "2
��u�R� @�u@��
+ �u(@�u
@�x� "2
���R� @
�h
@�x� @�u@��
)
�=
6
= �@�p
@�x+
2�
�2
�1
�� �R2� @@��
(��@�u
@��) +O("21)
�; (4)
@�p
@��= O("2"
21);
�vj��=1 =@�h
@�t; �uj��=1 = 0; �vj��=0 = 0
@�u
@��j��=0 = 0: (5)
If we neglect the O("21) terms, this is of course the system established by Lingand Atabek (1972) [16]. Nevertheless, we shall try to get rid of the simpli�cationthat they introduced in the expression of the longitudinal gradient: �u@�u@�x : Havingde�ned accurately the uid, we now look at the wall. As a complete descriptionis out of our scope, it will be much more simpli�ed than the uid.
3.3 Final form for the wall
It is diÆcult to model the artery and all the tissues around it. Often the arteryis described as an elastic pipe. For example, a simple shell description is usedin Belardinelli & Calvalcanti (1992) [2]. A constrained pipe description maybe found in Ma, Lee & Wu (1992) [17], in Kuiken (1984) [10] or in Pedley
(1980) [23]. This involves of course inertia of the wall, longitudinal tension @2�h@�x2
as well as circumferential tension, viscoelasticity, and careful treatment of theboundary conditions (note that inertia of the wall and nonlinearities may leadto solitary waves: Yomosa (1986) [42], Paquerot & Remoissenet (1994) [21]).Ohayon & Chadwick (1988) [20] noted that the pipe is composed of di�erentnon-homogenous media of distinct principal axis, and Teppaz et al (1995) [33]include a law in which the shear stress plays a key role. Viscoelasticity may beadded as in Horsten et al. (1989) [12], or in a shell theory as in Moodie et al.(1984) [19]. If all the non-local e�ects are neglected, a local law alone may beproposed. Di�erent local laws linking the pressure with the surface were testedby Tardy et al. (1991) [32], for example their best �t may be written as:
�(1 + "2h)2 = �(
�
2+ tan�1(
p0 + "2p
)): (6)
After inversion, one possible law is p = FC(h) depending on a set of parametersC = (�; ; p0). In this paper we use the most simple phenomenological non-linear law:
p = kh+ k2h2: (7)
We assume that the pipe is straight, without any bifurcation and that taperingis negligible. This last e�ect may be simply introduced by allowing a variatingR0:We may justify this choice if we claim that we study a prosthesis of a femoralartery. As our approach is simple, we may, in the future add other terms:
M@2h
@t2= (p� 2�
@v
@r)� FC(h) � s
@h
@t� S
@2h
@x2;
(with FC(h) = (kh + k2h2 + k3h
3:::); set of parameters C = (k; k2; k3) or ifFC(h) is equation (6) then C = (�; ; p0)) and construct the inverse method in
7
order to �nd the best set of parameters (C;M; s; S:::). These parameters mayeven depend on x (because of tapering etc). Note here, that having adimen-sionalized the perturbation of pressure by "2kR0, the pressure- displacementlaw (7) becomes �p = �h+ �k2�h
2: Nevertheless, as the elasticity will be one of theparameters which we shall look for, we shall write
�p = �k1�h+ �k2�h2; (8)
in the following �k1 = 1 (by construction); except in the last section where wewill construct a method to �nd exactly �k1 = 1. By de�nition the compliance is@S=@p, so with our notations:
@S=@p = (2�R0k�1)(
1 + "2�h
1 + (k2=(k"2))�h):
4 Integral equations
4.1 The equations
Here we adapt Von K�arm�an integral methods (from aerodynamics Schlichting(1987) [29]) to the system (3-4). The key is to integrate the equations withrespect to the variable � from the centre of the pipe to the wall (0 � �� � 1).So, we introduce �U0, the velocity along the axis of symmetry, a kind of loss of ux �q, and �� as follows:
�U0(�x; �t) = �u(�x; �� = 0; �t); �q = �R2( �U0�2Z 1
0�u��d��) & �� = �R2( �U2
0�2Z 1
0�u2��d��):
(9)We note that �q is like the ux di�erence between a perfect uid pro�le andthe real one; it is analogous to the displacement thickness Æ1 well known inaerodynamics. �� is nearly analogous to the energy displacement thickness Æ2.In aerodynamics the shape factorH links Æ1 and Æ2. Our new unknown functionsare q, R and U0, and we now establish their P.D.E. of evolution. Once againin establishing the uid motion equation, we suppose that "2 is not necessarilytoo small and � = O(1). The transverse integration of the incompressibilityrelation (3) with the help of the boundary conditions (5) gives:
@ �R2
@�t+ "2
@
@�x( �R2 �U0 � �q) = 0; �R = 1 + "2�h: (10)
If we integrate (4), with the help of the boundary conditions (5), we obtain theequation for q(x; t):
@�q
@�t+ "2(
@
@�x��� �U0
@
@�x�q) = �22�
�2�; � = (
@�u
@��)j��=1 � (
@2�u
@��2)j��=0: (11)
From the same equation (4) (and from (5)), evaluated on the axis of symmetry(in � = 0), we obtain an equation for the velocity along the axis U0(x; t):
@ �U0
@�t+ "2 �U0
@ �U0
@�x= �@�p
@�x+ 2
2�
�2
�0�R2; �0 = (
@2�u
@��2)j��=0: (12)
8
The two previous relations introduced the values of the friction in � = 0, the axisof symmetry: ((@
2�u@��2
)j��=0) and the skin friction in � = 1, at the wall: ((@�u@�� )j��=1).
Information has been lost here, so we need a closure relation between (��; �; �0)and (�q; �R; �U0). As there are so far no ambiguities, we remove the bars over theadimensionnalized symbols.
4.2 Closure
4.2.1 The selected velocity
As in aerodynamics, the previous system of equations is not closed: we havelost details of the velocity pro�le in the integration process. Therefore, wehave to imagine a velocity pro�le and deduce from it relations linking �, �and �0 and q, U0 et R. These relations are found from the radial dependenceof u. Pohlhausen's idea, explained in Schlichting (1987) [29] or Le Balleur(1982) [14], consists in postulating an ad hoc velocity distribution in � which�ts the boundary conditions and "looks like" observed pro�les. Here the mostsimple idea is to use the pro�les from the analytical linearized solution given byWomersley (1955) [38] for the case with no transverse pressure variation thatwe have already seen. This solution in complex form (i2 = �1) is rewritten as:
UWomersley = (FW (x; t) + iGW (x; t))(jr(��) + iji(��)); (13)
where FW ; GW ; ji and jr are real functions de�ned as follows:
(FW (x; t)+iGW (x; t)) =kp
c(1� 1
J0(i3=2�))ei2�(t�x=c); (jr+iji) =
0@1� J0(i3=2��)
J0(i3=2�)
1� 1J0(i3=2�)
1A :
Thus, we will assume that the velocity distribution in the following has thesame dependence on �. It means that we suppose that the fundamental modeimposes the radial structure of the ow. The real velocity is:
u = 1=2 ((F + iG)(jr + iji) + cc) = (Fjr �Gji); (14)
where F (x; t) and G(x; t) are now real unknown functions that we want to �ndand cc is the conjugate complex. We immediately see that U0(x; t) = F (x; t)(because jr(0) = 1 and ji(0) = 0) and that if we compute q with (14) we obtainG(x; t) as:
G(x; t) =q=R2 � U0 + U02
R 10 jr�d�
2R 10 ji�d�
: (15)
The two functions F and G are only functions of (U0; R; q) and we keep theWomersley radial dependence.
4.2.2 The coeÆcients of closure
The velocity at any radius � (14) and (15) may be written with the valueof the velocity at the centre U0 ,the radius R, and the loss of ux q: Next, by
9
integration, we obtain � as a function of (U0; R; q) and, by derivation, we obtain� and �0 as functions of (U0; R; q):
� = qqq2
R2+ quqU0+ uuR
2U20 ; � = �q
q
R2+�uU0 �0 = �0q
q
R2+�0uU0: (16)
This closes the problem. The coeÆcients (( qq; qu; uu); (�q; �u); (�0q; �0u)) areonly functions of �: They involve combinations of integrals and derivatives ofthe Bessel function. For example we have (if
Rf is a shorthand for
R 10 f(�)d�
and @�f�=0 an other for @f@� (0)):
uu = 1�Z
j2i =(
Zji)
2 � (2
Zjrji)=
Zji �
Zj2r +
+(2
Zj2i
Zjr)=(
Zji)
2 + (2
Zjijr
Zjr)=
Zji �
�(Z
j2i (
Zjr)
2)=(
Zji);
�0u = @2�jr�=0 + @2�ji�=0=
Zji � (@2�ji�=0
Zjr)=
Zji:
These coeÆcients are nearly constant for � < 5. For � small we obtain fromthe preceding computations:
((�65;11
5;�215
); (24;�12); (�12; 4)); (17)
so, we recover the values for the Poiseuille pro�le at small frequency. Thefact that those coeÆcients are nearly constant makes the model robust. For� ! 1 (in practice, � > 12 is enough) we �nd from asymptotic behaviour ofBessel functions and from the preceding computations the asymptotic form ofthe coeÆcients:
((��4p2; 2;�
p2
2�); (�2=2;��
p2); (0; 0)):
One can easily show that this is coherent with Wormesley's solution in the limitof large �. We note that for �!1 and "2 = 0, the wave solution for q is
q =
p2
��(1� i)e2i�(t�x=x); c =
sk
2(1�
p2
�(1� i) +O(��2):
Now equations (8), (10), (11) and (12 ) with the closure (16) de�ne a set of fourmonodimensional equations linking the pressure p, the velocity along the axisU0, the loss of ux q and the variation of the radius h.
4.2.3 Remarks
1- The main di�erence from other integral methods ([12], [18] , [23], [25], [27],[43], [44], or [41] ...) in our approach is the introduction of an auxillary partial
10
di�erential relation (11) obtained from an aeronautical analogy. Instead of q; �and U0 authors mainly use Q; Q2 and U0:
Q =
Z R
02�urdr Q=� = U0R
2 � q
Q2 =
Z R
02�ru2dr Q2=� = U2
0R2 � �
If we substract (10) from (11) we obtain the classical system of two equations:
2�R@R
@t+ "2
@
@x(Q) = 0;
@Q
@t+ "2
@
@x(Q2) = ��R2 @p
@x+ �
2�
�2(@u
@�)j�=1
Often, the relation forQ2 is written asQ2 =Q2
�R2 (in this case the radial variation
of the pro�le is neglected: at pro�le) or Q2 = 4Q2
3�R2 (parabolic pro�le: seeequation (17)). Note, that we have instead a third di�erential equation to linkQ1 and Q2. The e�ect of the skin friction (�1 =
2��2 (
@u@� )j�=1) is often estimated
by �1= �8��2
Q�R3 , true for a Poiseuille ow only ((17) again). It may be replaced
by an unsteady relation (deduced from unsteady Poiseuille ow) such as:
T�@�1@t
+ �1 = � 8
�2(Q+ TQ
@Q
@t+ :::)
See Yama et al (1995) [41] for the derivations and values of coeÆcients T� andTQ. We do not claim that our description is better, but for a sinusoidal inputwe �nd again (at any frequency) the Womersley linear solution. Our pro�lesare realistic in the sense that they present overshoots in the core and back ownear the wall. This is not the case when the closure is simply �1= �8�
�2Q�R3 or
in the case of very peculiar pro�les chosen by Belardinelli & Cavalcanti (1992)[2].
2- We noted that the coeÆcients vary little with �; this shows that our modelis very robust: it is easy to see that equations (10) and (11) are invariant underthe rescaling t! t=, =
p, and c ! c; if � is taken constant (independant of
�). This explains why methods based on Poiseuille coeÆcients are robust too.
5 Numerical resolution of the systems
5.1 Numerical resolution of the boundary layer system
Equations (3)- (5) are discretized in a simple way: a scheme implicit in (t; �) butexplicit in x, �rst order in time but second order in transversal and longitudinalvariables. The non linearity is handled by an internal loop between two timesteps (until the maximum of the radius amplitude's variation is smaller thana given small ", 10�5 in practice). The u@u@x term has been discretized with-out any hypothesis and induces no trouble. But, we must keep in mind that
11
boundary layer equations, in the case of a back ow, may lead to a singularity(Van Dommeln & Shen (1980) [34]) at a �nite time or may present instabilities(Cowley, Hocking & Tutty (1985) [5]). Nevertheless, no diÆculties were foundin the pulsated ow r�egime for which those equations were settled ("2 not toolarge, � of order 1 to 6).
The boundary condition at the input is only a given displacement h: ie weimpose h(xin; t): The discrete velocity pro�le u(xin; t) is obtained by a simplelinear extrapolation from the two next nodes as suggested by Hirsch (1990)[11]. The domain is long enough to avoid re exion at the output, or the outputh(xout; t) is given and the output velocity extrapolated as well.
5.2 Numerical resolution of the integral system
The integral system (10) - (11) is solved with simple techniques too. It has theform:
@f
@t= '
@F
@x+ �;
we code it with an Adams Bashford two step method. This scheme is secondorder in time and space. For the boundary conditions, we follow Hirsch (1990)[11], so we impose the displacement at the input and the output (h(xin; t) andh(xout; t)). The derivatives at the entrance are evaluated upstream and at theoutput downstream. The fact that the displacement is imposed at both endspermits output impedance problems to be to expelled.
6 Direct comparisons of the two codes
In this section, the input h(xin; t) is given, the output is far enough to avoidre exions during the time of computation. The parameters are �xed and arethe same for the two codes, k1 = 1 and k2 = 0. The codes were tested in thelinearized Womersley solution case ("2 = 0).
In �gure 2 we compare the models in the non-linear case (� = 3 and "2 6= 0).We observe the nonlinear sti�ening of the sinusoid. Increasing "2 and � maylead to a shock (Rudinger (1970) [28] or Cowley (1982) [4]). Our discretisationis not well adapted for shocks, but in rewriting it in a conservative way (witharti�cial viscosity) it should be possible to catch discontinuities.
To simulate clinical data by thin layer code (3)-(5) and (7), we put at theentrance a pseudo physiological law which is periodical in t of period 1:
hin(t) = 5te�100(t�1=6)2 + 0:5e�16(t�1=2)2 : (18)
One example of comparison is in �gure 3.
12
-0.3
-0.2
-0.1
0
0.1
0.2
h(x,
t=2.
5)
x
0.3
0.4
0.5
0.6
0.7
0.8
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
B.L.wom3(x,2.5)
intg.
Figure 2: The displacement of the wall (h(x; t = 2:5)) as a function of x isplotted here at time t = 2:5. The dashed line (wom3(x,2.5)) is the Womersleysolution (reference), the solid line (B.L.) is the result of the Boundary Layercode and the dots (intg) are the results of the integral method (� = 3, k1 = 1,k2 = 0 and "2 = 0:2).
13
0
0.05
0.1
0.15
0.2
h(x,
t=2)
0.25
0.3
0.35
0.4
0 0.2 0.4 0.6 0.8 1x
1.2 1.4 1.6 1.8 2
intgB.L.
Figu
re3:
Thedisp
lacementofthewall
(h(x;t=2))
asafunction
ofxisplotted
here
ata�xed
timet=2.
Theentran
ceisthetwobumpsfunction
.Thedash
edlin
e(B.L.)istheresu
ltof
theBoundary
Layer
codeandthesolid
line(in
tg)is
theresu
ltof
theintegral
meth
od(�
=4,k1=1,k2=0and"2=0:1).
14
7 Comparisons of the two resolutions: the inverse
method
7.1 Scope of the method
In the preceding sections, we have constructed two models for wave propagation,the second one being a more simpli�ed version of the �rst, which is itself a hugesimpli�cation of the physical problem. The purpose of this paper is to constructan inverse method which will allow the evaluation of the best set of parametersfor a model in order to �t experimental data obtained in a non-intrusive way.We suppose that we are able to measure the displacement of the artery at threedistinct locations: xin, xm and xout (say we have hin(t), hm(t) and hout(t)). Thecomputational domain will be [xin, xout], and the measured values are boundaryconditions for the computational model: h(xin; t) = hin(t) and h(xout; t) =hout(t). Here, we want to �nd the values of k1k2 and � that give the bestagreement between h(xm; t) and hm(t). As we do not have experimental data,the time series hin(t), hm(t) and hout(t) will be produced by the thin layer code.
Following Chavent (1979) [3], and taking distances with careful mathemat-ical de�nitions of the functional spaces, we now build the adjoint problem inorder to minimise a cost function. First, we have to de�ne this cost functionas an integral criterion which cumulates the errors between h(xm; t) and hm(t)during a time period (of length 1):
J =
Z 1
0(h(xm; t)� hm(t))
2dt: (19)
This criterion must be as small as possible. Next, we de�ne a variational for-mulation of our problem: if we denote (10) as Eh = 0, (12) as Eu = 0 and (11)as Eq = 0, then for any test function h� u� and q� in the ad hoc space:
E =
Z xout
xin
dx(Ehh�) +
Z xout
xin
dx(Euu�) +
Z xout
xin
dx(Eqq�) (20)
is zero for U0 q and h solutions of the integral problem (10) - (11) and (8). Oncethe variational formulation has been de�ned, the Lagrangian of the optimisationproblem is de�ned by adding criterion (19) to the variational formulation (20)integrated over a time period:
L =
Z 1
0dt(E) + J: (21)
The equations (Eh, Eu and Eq) are here the constraints and (h� u� and q�) arethe Lagrangian multiplicators. For any solution (h(�; k1; k2); U0(�; k1; k2); q(�; k1; k2))of (10) -(11) and (8) E is zero,
E(h(�; k1; k2); U0(�; k1; k2); q(�; k1; k2)) = 0
and the Lagrangain reduces to the criterion:
L(h(�; k1; k2); U0(�; k1; k2); q(�; k1; k2)) = 0+J(h(�; k1; k2); U0(�; k1; k2); q(�; k1; k2)):
15
By di�erentiating this last equation we obtain:
ÆJ =@L
@hÆh+
@L
@qÆq +
@L
@U0ÆU0 +
@L
@��+
@L
@k1Æk1 +
@L
@k2Æk2: (22)
We choose at this point the adjoint variables h� u� and q� such that:
@L
@h= 0;
@L
@q= 0 &
@L
@U0= 0: (23)
This is the adjoint problem that we have to solve. Then equation (22) becomes:
ÆJ =@L
@�� +
@L
@k1Æk1 +
@L
@k2Æk2: (24)
this is the sought expression for the gradient: it depends as well on the direct(10)-(12) as on the adjoint (25)-(27) problem.
During manipulations of L (22) and (23) the trick is to make integrationsby parts, for example terms like
R Rdtdx(@Æh@t h
�) are changed intoZ Zdtdx(�@h
�
@tÆh) +
Zdx[Æhh�]t=1
t=0;
note that the new system is backward in time (�@h�
@t ). As there is no error inÆh at t = 0, this allows us to �x in a natural way the inital boundary condi-tions for the back propagation problem such as h�(x; t = 1) = 0. The samemanipulations are done for the spatial derivations leading to the boundary con-ditions at the edges. Also note when estimating ÆL we obtain a source term inthe equation of h� from the cost function di�erential Æ
R 10 (h(xm; t)� hm(t))
2dt,
that we write as:R 10
R xoutxin
Æh ( 2(h(x; t) � hm(t))Æxmdxdt), where Æxm is the Diracdistribution at point xm.
7.2 The �nal adjoint system
If we apply the above method with equations (8), (10) -(11) and (23),we obtainthe following P.D.E., which is the �nal adjoint system:
�2R0@h�
@t� k1
@u�
@x� k2h
@u�
@x+ 2(h� hm)Æxm = 0 (25)
�@u�
@t�R2
0
@
@x(h�)� 2 2�
�2R20
�0uu� +
2 2�
�2(�uq
�) = 0 (26)
�@q�
@t+
@
@x(h�)� 2 2�
�2R20
(�0q1
R20
u�) +2 2�
�2(�q(
1
R20
)q�) = 0 (27)
The boundary conditions are for any x : u�(x; t = 1) = 0; h�(x; t = 1) = 0;q�(x; t = 1) = 0; and h�(xin; t) = 0 and h�(xout; t) = 0. The �nal threecomponents of the gradient (22) of the cost function are:
16
@
@�J =
ZZdtdx(� @
@�(2 2�
�2R2�0q)
q
R2u� �
� @
@�(2 2�
�2R2�0u)U0u
� +@
@�(2 2�
�2�q)
q
R2q� +
+@
@�(2 2�
�2�u)U0q
�) (28)
and
@
@k 1J =
ZZdtdx(u�
@h
@x)
@
@k 2J =
ZZdtdx(u�
@h2
@x) (29)
To check the system (25) -(27) we have found a solution for it in terms of amoving plane linear wave e�2i�(t�x=ci). We �nd that the complex phase velocityci is the complex conjugate of the Womersley phase velocity.
7.3 Numerical discretisation
The adjoint system (25)-(27) is solved using the Adams Bashford method. TheDirac distribution is approximated by a Gaussian function of standard deviationequal to the step size (�x = :01).
A loop of the computation of the gradient is then as follows: �rst, the inputand output come from the thin layer computation (3)-(5), and starting fromt = 0, U0 = q = h = 0; we solve (10) -(11) and (8) with a given set (�; k1; k2),and reiterate until the forced r�egime is settled (i :e: the di�erence between twoperiods is less on average than 5:10�5, which is achieved after 6 or 7 periods).All the values during the next period are stored for the gradient's computation.Second, a back propagation computation of (25)- (27) is performed: the com-ponents of the gradient of J (28) and (29) are the results. Third, the valuesof (�; k1; k2) are updated into new ones. For this updating, we tested a simpleconstant step gradient method and a more sophisticated technique from Presset al. (1995) [24]. This new set is used for the next loop until convergence:i :e: when we approach the minimum of the cost function. The simple constantstep method was found to be the more robust. Of course, a great number ofiterations is necessary to obtain the minimum.
7.4 Results
7.4.1 Generation of data
First, we run the boundary layer code (3)-(5), with an appropriate entry (seebelow), the output is in 1 where we put h(1; t) = 0. This allows re exions(any other h(1; t) may be imposed, it does not change the result). The input(h(xm; t)) and two measurements (h(xm; t) and h(xout ; t)) are stored after aforced r�egime has taken place (t > 4). In practice xin = 0; and the two valuesare xm = 0:12 and xout = 0:2: Of course, particuliar values for the viscosity andthe wall coeÆcients have been used. Second, we run the backpropagation codeto try to retrieve the original values.
17
7.4.2 Linear wall case with sinusoidal entry
In this full linearized case, (k2 is imposed to be 0, so we �rst guess � and k1) we
use at the entry hin(t) = sin(2�t). This is the Womersley problem. About 300iterations are necessary to obtain the set of parameters. In the range 3 < � < 6and 0:7 < k1 < 2, the di�erence between the initially given value and theguessed value is at most 1:5%, the value of k is precise to 0:5%. The agreement,with regard to the numerical errors, is excellent.
7.4.3 Non-linear wall case with sinusoidal entry
In this linearized uid case ("2 = 0) we allow the wall to be non linear k2 6= 0.Three examples are computed here corresponding to (� = 5; k1 = 1; and k2 =0), (� = 5; k1 = 1; and k2 = 0:1) and (� = 5; k1 = 1:25; and k2 = 0:2). In �gure4, 5 and 6 we display respectively the �; k1 and k2 history versus the numberof iterations.
This process is slower (it takes about 1500 iterations to obtain the threeparameters). We note that the introduction (in Womersley case) of the newparameter does not a�ect the precision of � (1.5%) or k1 (0.4%), the �nal valueof k2 is 5:10
�3 (instead of 0). So the agreement is good. If we increase the valueof k2 to 0:1 the error is about 2% for � and less than 1:% for k1 and the erroron k2 is approximately 0:04. Increasing k2 to 0:2 gives an error of 5% on �, 5%on k1 and again the error on k2 is approximately 0:04. Within the range of theparameters, the absolute error is about 0:2 for � and 0:05 for k1 and k2:
7.4.4 In uence of the noise
A simple investigation has been carried out on the in uence of noise on thedata, a random value lying in the interval [-"n; "n] is added to hin(t), hm(t)and hout(t) in the pure Womersley case. The result of several computations("n = 0:2) on the guessed values (�; k1; k2) shows that the number of iterationsis again nearly around 1500. The absolute error is about 0:25 for �, 0:02 for k1and 0:01 for k2.
7.4.5 linear case physiological
Finally we put at the entry the pseudo physiological law (18). In the linear uidcase "2 = 0, the di�erence on the value of � is about 2.5% and on k about 7:5%:Then if we increase the non-linearities to "2 = 0:05 (respectively "2 = 0:1), theirin uence on the result is an error of 13% for � (resp. 24%). In �gure 7 and 8we see an example of increasing of "2.
8 Conclusion
We have presented here a numerical resolution of a set of simpli�ed equationsissued from Navier Stokes equation (Ling & Atabek (1972) [16]). A simpli�edintegral method, slightly di�erent from the preceding ones, has been presentedtoo. The two methods work fairly well if viscosity or non-linearity are changed
18
3.8
4
4.2
4.4
4.6
4.8
5
5.2
0 500 1000 1500 2000 2500
’a=5. k1=1.00 k2=0.0’’a=5. k1=1.00 k2=0.1’’a=5. k1=1.25 k2=0.2’
5
Figu
re4:
Variation
of�in
thethree
casesas
afunction
ofthenumber
ofiteration
s.
19
0.95
1
1.05
1.1
1.15
1.2
1.25
1.3
0 500 1000 1500 2000 2500
’a=5. k1=1.00 k2=0.0’’a=5. k1=1.00 k2=0.1’’a=5. k1=1.25 k2=0.2’
11.25
Figu
re5:
Variation
ofk1in
thethree
casesas
afunction
ofthenumber
ofiteration
s.
20
-0.05
0
0.05
0.1
0.15
0.2
0.25
0 500 1000 1500 2000 2500
’a=5. k1=1.00 k2=0.0’’a=5. k1=1.00 k2=0.1’’a=5. k1=1.25 k2=0.2’
.1
.2
Figu
re6:
Variation
ofk2in
thethree
casesas
afunction
ofthenumber
ofiteration
s.
21
2.8
3
3.2
3.4
3.6
3.8
4
4.2
4.4
4.6
4.8
5
0 200 400 600 800 1000 1200 1400 1600 1800 2000
e2=0.00e2=0.05e2=0.10
4
Figu
re7:
Variation
of�(sim
ulation
of�
=4,
k1=
1:2andk2=
0.and
"2=0:1)
with
"2=0,"2=0:05
and"2=0:1
22
1.15
1.2
1.25
1.3
1.35
1.4
1.45
1.5
1.55
1.6
1.65
1.7
0 200 400 600 800 1000 1200 1400 1600 1800 2000
e2=0.00e2=0.05e2=0.10
1.2
Figu
re8:
Variation
ofk1(sim
ulation
of�=
4,k1=
1:2andk2=
0.and
"2=0:1)
with
"2=0,"2=0:05
and"2=0:1.
23
(other comparisons should be done using the analytical results from Wang &Tarbel (1992) [36] or Wang & Tarbel (1995) [37]). A better comparison has beendone in constructing an inverse method: the data from the �rst code (�; k1; k2)were found by the integral method after the resolution of a backpropagationsystem. Encouraging results are found, even when a small amount of noise isadded. This inverse method may be adapted to other sets of simple integralequations and non-linear e�ects in the uid may be, in principle, easily added inthe description (but will require more computer time anticipating the increaseof longitudinal resolution). Of course an inverse method based on (3)-(5) maybe built (anticipating that the power of computers increases). Maybe the resultsobtained are too simple in comparison to the relative complexity of the inversemethod: but while a two parameters try/error shot (only � k1 with k2 = 0)may be settled without the help of the inverse method, a try/ error procedurewith 3 parameters is too diÆcult. The principles of the method were settled forfuture use: the challenge is now to use real data, obtained by a non-intrusiveway, (or numerical data issued from other numerical models) to evaluate mainlythe elasticity of the wall. In fact any other phenomena, like a variating k, or avariating R0 may be added too, opening the way to the detection of stenosis oraneurysm.
AknowledgementsThe author would like to thank Maurice Rossi (LMM) for discussions. Thiswork was supported by the Direction Sup�erieure des Programmes TechniquesD.S.P.T. 8.
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[45]
[46]
[47]
[48] % FICHIER "C601ter:tex"
27
